Multiplying Complex Numbers

The multiplication of complex numbers involves finding the product of two or more complex numbers using the distributive property.

That is, for two complex numbers: \(z_1 = a + ib \) and \(z_2 = c + id \), the product of z₁ and z₂ is \(z_1 \times z_2 = (a + ib)(c + id) \).

The multiplication of complex numbers follows similar properties to the real numbers. The properties are 

• Commutative property
   
• Associative property
   
• Distributive property

Commutative property: The order of multiplication does not affect the result. For example, \(z_1 \times z_2 = z_2 \times z_1 \) 

Associative property: The associative property of complex numbers states that the order of grouping the complex numbers doesn't change the result. That is, \(z_1 \times (z_2 \times z_3) = (z_1 \times z_2) \times z_3 \) 

Distributive property: The distributive property states that z₁ \(z_1 \times (z_2 + z_3) = z_1 z_2 + z_1 z_3 \)

Now, let’s learn how to multiply complex numbers in Cartesian form. In this form, we multiply the complex number term by term. 
 

That is, if\(z_1 = a + ib \) and \(z_2 = c + id \)

Then \(z_1 \times z_2 = (a + ib)(c + id) \)

= \(ac + a (id) + ib (c) + i^2 bd \)

= \((ac - bd) + i(ad + bc) \)

So, \((a + ib)(c + id) = (ac - bd) + i(ad + bc) \)

For example, find the product of \(z_1 = 3 + 2i \) and \(z_2 = 1 + 4i \)
z₁z₂ = (3 + 2i) (1 + 4i)

Using the formula for multiplying complex numbers, \((a + ib)(c + id) = (ac - bd) + i(ad + bc) \)

\((3 + 2i)(1 + 4i) = ((3 \times 1) - (2 \times 4)) + i((3 \times 4) + (2 \times 1)) \)

=\((3 - 8) + i (12 + 2) \)

= \(-5 + 14i\)

When multiplying complex numbers in polar form, we first multiply the moduli and then add the arguments. That is, when multiplying \(z_1 = r_1 e^{i\Theta_1} \) and \(z_2 \)= \(r_2 e^{i \Theta_2} \)
\(z_1 \times z_2 = r_1 r_2 e^{i(\Theta_1 + \Theta_2)} \)

For example, multiplying \(z_1 = 2 e^{i \pi/6} \) and \(z_2 = 3 e^{i \pi/4} \)

\(z_1 z_2 = r_1 r_2 e^{i(\Theta_1 + \Theta_2)} \)

Here, r₁ = 2

r₂ = 3

Θ₁ = \(\frac{\pi}{6} \) 

Θ₂ = π/4

Multiplying moduli \((r_1 r_2) = 2 \times 3 = 6 \)

Adding the arguments = \(\frac{\pi}{6} + \frac{\pi}{4} \)

= \(\frac{2\pi}{12} + \frac{3\pi}{12} \)

= \(\frac{5\pi}{12} \)
 

So, \(2 e^{i \pi/6} \times 3 e^{i \pi/4} = 6 e^{i 5\pi/12} \)

The multiplication formula for complex numbers is  \((a + ib)(c + id) = (ac - bd) + i(ad + bc) \). When multiplying the complex number by a real number, let's consider b as 0. That is \(a\) \((c + id) = ac + i a d \). 

For example: \(-5i (2 + 3i) \)

Using the formula, \(a (c + id) = ac + i a d \)
Here, a = -5i

c = 2

d = 3i

Substituting the value, \(-5i (2 + 3i) = (-5i \times 2) + (-5i \times 3i) = -10i + (-15 i^2) \)

Substituting \(i^2 = -1 \)

=\(-10i + 15 = 15 - 10i \). 

The formula for multiplying complex numbers is \((a + ib)(c + id) = (ac - bd) + i(ad + bc) \). When squaring a complex number, consider a = c and b = d, and then the formula becomes;

\((a + ib)^2 = (a \times a - b \times b) + i(ab + ba) \)

\((a^2 - b^2) + i (2ab) \)

For example, squaring \(4 + 5i \)

\((a + ib)^2 = (a^2 - b^2) + i(2ab) \)

So, \((4 + 5i)^2 = (4^2 - (5i)^2) + i(2 \times 4 \times 5) \)

= \((16 - 25 \times i^2) + 40i \)

= \(16 - 25 + 40i \)

= \(-9 + 40i \)

The multiplicative inverse of a complex number \(z = a + ib\) is another complex number \(z^{-1} \), that is, \(z \times z^{-1} = 1 \). For a complex number \(z = a + ib \), the multiple inverse \(z^{-1} \) = \( \frac {\bar z} {|z|^2}\)

Here,\(z = a - ib \), so, \(|z| = \sqrt{a^2 + b^2} \)

The formula for the multiplicative inverse: \(z^{-1} \) = \( \frac {\bar z} {|z|^2}\)

Where, conjugate:\(z = a - ib \), and the modulus of the complex number is \(|z| = \sqrt{a^2 + b^2} \)

Learn effective strategies to multiply complex numbers accurately, both in standard and polar forms, and understand their geometric interpretations.

• Always use the formula \((a + ib)(c + id) = (ac - bd) + i(ad + bc) \).
   
• Separate real and imaginary parts before multiplying.
   
• Practice squaring complex numbers to see patterns.
   
• Use the Argand plane to visualize multiplication geometrically.
   
• Convert to polar form to simplify multiplication: \(r_1 e^{i\Theta_1} \cdot r_2 e^{i\Theta_2} = r_1 r_2 e^{i(\Theta_1 + \Theta_2)} \).

Complex number multiplication is used in various fields, like engineering, quantum mechanics, computer graphics, etc. In this section, we will discuss them in detail. 

• Electrical engineering - Multiplying complex numbers is used to calculate impedance, voltage, and current in alternating current (AC) circuits, where signals have both magnitude and phase.
   
• Signal Processing - Complex number multiplication helps in analyzing and manipulating signals in communication systems, such as modulation and filtering.
   
• Control Systems - Engineers use complex numbers to represent system dynamics; multiplying them helps in understanding system stability and response.
   
• Quantum mechanics - Complex numbers represent quantum states, and their multiplication is used in computing probability amplitudes and transformations.
   
• Rotations in 2D space - Multiplying complex numbers is used to rotate vectors or points in a plane, which is useful in computer graphics, robotics, and navigation.